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$q$-Painlevé equations on cluster Poisson varieties via toric geometry

Yuma Mizuno

Abstract

We provide a relation between the geometric framework for $q$-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with $q$-Painlevé equations. We introduce the notion of seeds of $q$-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of $q$-Painlevé equations given by Sakai. We realize $q$-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of $q$-Painlevé type.

$q$-Painlevé equations on cluster Poisson varieties via toric geometry

Abstract

We provide a relation between the geometric framework for -Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with -Painlevé equations. We introduce the notion of seeds of -Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of -Painlevé equations given by Sakai. We realize -Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of -Painlevé type.

Paper Structure

This paper contains 8 sections, 34 theorems, 135 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on cluster varieties
  3. q-Painlevé systems on cluster Poisson varieties
  4. Seeds in lattices of rank 2
  5. Seeds of q-Painlevé type
  6. Fano polygons and the classification of seeds of q-Painlevé type
  7. Actions of Cremona isometries on cluster Poisson varieties
  8. Cluster data for q-Painlevé equations

Key Result

Theorem 1.1

The seeds of $q$-Painlevé type (Definition def:q-P type) modulo mutation equivalence are classified into the ten seeds in Figure fig:seeds of q-P type.

Figures (3)

  • Figure 1: Representatives of the mutation equivalence classes of seeds of $q$-Painlevé type. The numbers of marks at the end of each vector stand for the multiplicity of the vector in the seed. The symbol at the left of each seed is the symmetry type $R^{\perp}$ of the seed.
  • Figure 2: $R$
  • Figure 4: Representatives of the mutation equivalence classes of Fano polygons that have no remainders. These Fano polygons correspond the seeds in Figure \ref{['fig:seeds of q-P type']} after rotating $-\pi/2$.

Theorems & Definitions (81)

  • Theorem 1.1: See Theorem \ref{['thm:seeds are q-P']} and \ref{['thm:classification of q-P seeds']}
  • Theorem 1.2: See Theorem \ref{['theorem:qP action on X']}
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 71 more